Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Wolfram`QuantumFramework`
QuantumWignerTransform [ ]  |
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Details and Options
Examples
(26)
Basic Examples
(5)
Generate the Wigner transformation a quantum state in the phase space:
In[1]:=
ρ=
["RandomMixed",3];
[ρ]
 | QuantumState |
 | QuantumWignerTransform |
Out[2]=
QuantumState
 |  |
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It is the same as the basis transformation of the doubled state into the corresponding Wigner basis:
 | QuantumState |
 | QuantumBasis |
True
For odd dimensions, PhaseSpace as a property of a quantum state has the dimension :
d×d
In[1]:=
ρ=
["RandomMixed",3];MatrixForm[ρ["PhaseSpace"]]
| QuantumState |
Out[1]=
0.108242 | 0.163752 | 0.200072 |
0.16736 | 0.193372 | -0.026586 |
0.193771 | -0.02127 | 0.0212874 |
The state vector in the Wigner basis corresponds to PhaseSpace representation of the state:
In[2]:=
| QuantumWignerTransform |
Out[2]=
True
For even dimensions, PhaseSpace has the dimension :
2d×2d
In[1]:=
d=4;ρ=
["RandomMixed",d];ρ["PhaseSpace"]
| QuantumState |
Out[1]=
SparseArray
|  |
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This implies that the PhaseSpace of even dimensions has some extra dependent elements. It has been discussed in more details later.
If we partition the PhaseSpace into four blocks of the dimension , the upper left one corresponds to the StateVector of the Wigner transformation:
d×d
In[2]:=
| QuantumWignerTransform |
Out[2]=
True
Winger transformation of an operator:
In[1]:=
op=
["X"]
| QuantumWignerTransform |
| QuantumOperator |
Out[1]=
QuantumOperator
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Show its representation in the phase space:
In[2]:=
TraditionalForm[op]
Out[2]=
+--
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Wigner transformation of a quantum circuit (pay attention to wire dimensions):
In[1]:=
qc=
["Bell"];qc["Diagram","ShowWireDimensions"True]
| QuantumWignerTransform |
| QuantumCircuitOperator |
Out[1]=
Return the result of circuit on a register state:
In[2]:=
qc[]["Amplitudes"]
Out[2]=
,0,0,0,0,,0,0,0,0,,0,0,0,0,-
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16
Check that the Weyl transformation of the outcome is the same as the Bell state:
In[3]:=
QuantumWeylTransform[qc[]]["Formula"]
Out[3]=
1
2
1
2
Scope
(9)
Generalizations & Extensions
(4)
Applications
(6)
Interactive Examples
(2)
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